Norming Approximate Orthogonality in Normed Linear Spaces

We introduce and study the notion of \emph{norming approximate orthogonality}, a two-parameter generalization of Birkhoff--James orthogonality in normed linear spaces. For $δ, \varepsilon \in [0,1)$ with $\varepsilon < (1-δ)^2$, we say $x \nperp y$ in $X$ if there exists $f \in X^*$ with $|f(x)| \ge